In a large bowl, add the flour, salt and butter. Wearing disposable gloves, we begin to rub the butter with the flour with our fingers, until the mixture looks like crumbs.: The air is cool when you begin. As you rub the cold butter into the all purpose flour , you should hear a faint whisper as the crumbs form, and the mixture will feel sandy between your fingers. This technique creates little pockets of butter that melt during baking and give the crust layers, so do not overwork the dough or the texture will become dense. A common mistake is warming the butter with your hands, which makes the dough greasy, so work briskly and consider chilling the bowl if your kitchen is warm.
Add the water, 1 egg yolk and knead until you create a smooth dough.: After adding the cold water and one egg yolk , press the mixture together with fingertips or a bench scraper; the dough should come together without being sticky. You will feel it transform from crumbly to cohesive, and the surface should be smooth. Knead just enough to bind; over kneading tightens gluten and yields a tough crust. If the dough feels sticky, dust lightly with flour rather than adding more water.
Shape the dough into a disc, wrap it in plastic wrap and put it in the refrigerator for 1 hour.: Chilling firms the butter and relaxes the gluten so the dough rolls easily and bakes flakier. The dough will feel cold and slightly glossy after chilling. Resist the urge to shorten this rest, because a warm dough will shrink when baked. If you skip chilling you risk a crust that warps or contracts in the oven.
Preheat the oven to 160°C (320°F) in fan mode.: The oven should steadily come to temperature, producing a dry heat that encourages even crust baking. You may notice a subtle hum as the fan circulates air, which helps the pastry brown evenly. If your oven runs hot, use an oven thermometer to avoid an overbrowned base. Preheating fully is essential; placing pastry into an oven that has not stabilized can result in an underbaked interior.
Place 2 sheets of parchment paper on a workbench and sprinkle with flour. Using a rolling pin, roll out the dough between the parchment paper sheets. If the dough sticks, sprinkle it with a little more flour.: Rolling between parchment makes the process smooth and prevents sticking, and the extra flour on the sheets creates a buffer so the dough moves without tearing. The dough should roll out into an even circle, about the thickness of a coin, and if it clings slightly, a light dusting of flour will free it. Avoid excess flour which will dry the edges; a gentle touch keeps the pastry tender.
Grease with butter and flour a round tart pan, 26 cm (10 inches) in diameter, with a removable bottom. Spread the dough into the tart pan, pressing it well, so that it sticks to the bottom and sides of the pan.: Greasing ensures the baked tart releases cleanly, and the removable bottom helps when unmolding. Press the dough into the pan, making sure it adheres to the sides without stretching, which would cause shrinkage. Run a rolling pin across the top to remove excess. A common error is pressing too hard and compressing the layers, so aim for even pressure and a snug fit.
Pierce the bottom of the tart with a fork in many places. Cover the dough with plastic wrap and put some weight (beans, lentils or baking weights) on it, so that the dough does not rise during baking.: Docking the base prevents bubbles from forming, keeping the surface flat for filling. After covering with plastic and adding weights, the crust will bake without puffing, yielding a uniform shell. If you skip docking, you might end up with uneven pockets that make filling difficult and can cause leaks during blind baking.
Bake for 30 minutes and remove the tart base from the oven and brush it with the yolk spread. Raise the temperature to 180°C (360°F), bake again until golden and set the base aside.: As the crust bakes, you will notice it go from pale to a tender golden hue; the aromas will hint at toasty butter and baked flour. Brushing with the egg yolk wash at this stage seals the surface so it resists sogginess when the custard is added. Overbaking at this step will darken the crust too much, while underbaking leaves it fragile and prone to collapsing when filled.
Lower the oven temperature to 160°C (320°F) in the fan mode.: Reducing the heat prepares the oven for a gentler bake of the filled tart, promoting even set of the custard without overbrowning. You will switch from high initial heat to a steady, controlled environment where the filling can cook slowly. Skipping this adjustment risks a browned top before the interior is fully set.
Place a small pot, with enough salted water, over high heat. Once the water starts to boil, add the broccoli and boil for about 10 minutes. Remove with a slotted spoon and move to a bowl with water and ice.: When the water reaches a rolling boil, add the broccoli florets and cook until they are tender yet bright green. The boiling sound intensifies as you add the vegetables, and the blanching time is short so check often. Immediately shock the broccoli in ice water to stop cooking and preserve color. Overcooking here leads to limp, mushy florets that release water into the tart and dilute the flavor.
In another bowl, add the milk, cream, eggs, parmesan cheese, graviera cheese, feta cheese, spearmint, paprika and thyme. Season with salt and pepper and mix very well with a hand whisk, until the mixture is even.: Whisk until the mixture is smooth and glossy, and you will see the cheeses start to meld into the dairy. The aroma will become a layered savory scent with herbaceous notes, and the mixture should taste slightly seasoned but not overpoweringly salty. If the mixture tastes too bland, a touch more salt will help, but be cautious because the cheeses add salt naturally.
Spread the broccoli and mushrooms over the base of the tart, place it in the oven and carefully pour the filling mixture.: Arrange the blanched broccoli and cut white mushrooms evenly so each slice gets a balance of vegetables. When you pour the filling over them, watch as it seeps into crevices, surrounding the vegetables without displacing them. Press down lightly if large air pockets appear, but avoid packing them in which can compress the filling unevenly. Uneven distribution causes some slices to be too wet or too sparse.
Bake the tart for 1-1 1/2 hours, until golden. Set it aside to cool and serve.: The oven will gently hum as the custard sets; you will see the top turn a warm golden brown and the center should be just set with a slight wobble. The aroma at this stage is rich and deeply savory. Let the tart cool so it firms before slicing; cutting too soon will cause the filling to spill. A frequent mistake is removing it while the center is still liquid, so err on the side of a few extra minutes if uncertain.